An Evaluation of Error Variance Bias in Spatial Designs

  • Emlyn R. WilliamsEmail author
  • Hans-Peter Piepho


Spatial design and analysis are widely used, particularly in field experimentation. However, it is often the case that spatial analysis does not significantly enhance more traditional approaches such as row–column analysis. It is then of interest to gauge the degree of error variance bias that accrues when a spatially designed experiment is analysed as a row–column design. This paper uses uniformity data to study error variance bias in \(7\times 12\) spatial designs for 21 treatments.


Experimental design Row–column design Latin square Spatial design Linear variance Average efficiency factor Randomization 



We would like to thank two anonymous reviewers for their careful reading and constructive comments on an earlier version of this paper.


  1. Bailey, R. A. (2012), “Experiments in rectangular areas: design and randomization”, Journal of Agricultural, Biological and Environmental Statistics, 17, 176–191.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bailey, R. A., and Rowley, C. A. (1987), “Valid randomization”, Proceedings of the Royal Society of London, Series A, 410, 105–124.CrossRefzbMATHGoogle Scholar
  3. Coombes, N. (2002), “The reactive Tabu search for efficient correlated experimental designs”, PhD thesis, Liverpool, U.K.: Liverpool John Moores University.Google Scholar
  4. Cox, D. R. (2009), “Randomization in the design of experiments”, International Statistical Review, 77, 415–429.CrossRefGoogle Scholar
  5. Cullis, B. R., Smith, A. B., and Coombes, N. E. (2006), “On the design of early generation variety trials with correlated data”, Journal of Agricultural, Biological and Environmental Statistics, 11, 381–393.CrossRefGoogle Scholar
  6. Eccleston, J., and Chan, B. (1998), “Design algorithms for correlated data”, Compstat Proceedings, 13, 41–52.CrossRefzbMATHGoogle Scholar
  7. Fisher, R. A. (1925), “Statistical Methods for Research Workers”, Edinburgh, Oliver and Boyd.zbMATHGoogle Scholar
  8. ——– (1926), “The arrangement of field experiments”, Journal of the Ministry of Agriculture, 33, 503–513.Google Scholar
  9. Forkman, J. (2016), “A comparison of super-valid restricted and row-column randomization”, Journal of Agricultural, Biological and Environmental Statistics, 21, 243–260.MathSciNetCrossRefzbMATHGoogle Scholar
  10. Grundy, P. M., and Healy, M. J. R. (1950), “Restricted randomization and quasi-Latin squares”, Journal of the Royal Statistical Society, Series B, 12, 286–291.zbMATHGoogle Scholar
  11. Mead, R., Gilmour, S. G., and Mead, A. (2012), “Statistical Principles for the Design of Experiments”, Cambridge University Press.Google Scholar
  12. Monod, H., Azais, J. M., and Bailey, R. A. (1996), “Valid randomisation for the first difference analysis”, Australian Journal of Statistics, 38, 91–106.MathSciNetCrossRefzbMATHGoogle Scholar
  13. Muller, B. U., Kleinknecht, K., Mohring, J., and Piepho, H. P. (2010), “Comparison of spatial models for sugar beet and barley trials”, Crop Science, 50, 794–802.Google Scholar
  14. Nelder, J. A. (1965), “The analysis of randomized experiments with orthogonal block structure”, Proceedings of the Royal Society of London, Series A, 273, 147–178.CrossRefzbMATHGoogle Scholar
  15. Papadakis, J. S. (1937), “Méthode statistique pour des expériences sur champ”, Bull.Inst. Amel. Plantes à Salonique, 23.Google Scholar
  16. Piepho, H. P., Mohring, J., Pflugfelder, M., Hermann, W., and Williams, E.R. (2015), “Problems in parameter estimation for power and AR(1) models of spatial correlation in designed field experiments”, Communications in Biometry and Crop Science, 10, 3–16.Google Scholar
  17. Piepho, H. P., and Williams, E. R. (2010), “Linear variance models for plant breeding trials”, Plant Breeding, 129, 1–8.CrossRefGoogle Scholar
  18. Piepho, H. P., Williams, E. R., and Michel, V. (2016), “Nonresolvable row-column designs with an even distribution of treatment replications”, Journal of Agricultural, Biological and Environmental Statistics, 21, 227–242.MathSciNetCrossRefzbMATHGoogle Scholar
  19. Speed, T. P. (1990), “Introduction to ‘The arrangement of field experiments‘ by R.A. Fisher”, Technical Report No. 253, Department of Statistics, University of California, Berkeley.Google Scholar
  20. Speed, T. P., Williams, E. R., and Patterson, H. D. (1985), “A note on the analysis of resolvable block designs”, Journal of the Royal Statistical Society, Series B, 47, 357–361.MathSciNetzbMATHGoogle Scholar
  21. Stefanova, K. T., Smith, A. B., and Cullis, B. R. (2009), “Enhanced diagnostics for the spatial analysis of field trials”, Journal of Agricultural, Biological and Environmental Statistics, 14, 392–410.MathSciNetCrossRefzbMATHGoogle Scholar
  22. Tedin, O. (1931), “The influence of systematic plot arrangement upon the estimate of error in field experiments”, Journal of Agricultural Science, 11, 191–208.CrossRefGoogle Scholar
  23. Wilkinson, G. N., Eckert, S. R., Hancock, T. W., and Mayo, O. (1983), “Nearest neighbour (NN) analysis of field experiments (with discussion)”, Journal of the Royal Statistical Society, Series B, 45, 157–211.zbMATHGoogle Scholar
  24. Williams, E. R., and Luckett, D. J. (1988), “The use of uniformity data in the design and analysis of cotton and barley variety trials”, Australian Journal of Agricultural Research, 39, 339–350.CrossRefGoogle Scholar
  25. Williams, E. R., John, J.A., and Whitaker, D. (2006), “Construction of resolvable spatial row-column designs”, Biometrics, 62, 103–108.MathSciNetCrossRefzbMATHGoogle Scholar
  26. Williams, E. R., and Piepho, H. P. (2013), “A comparison of spatial designs for field trials”, Australian and New Zealand Journal of Statistics, 55, 253–258.MathSciNetCrossRefzbMATHGoogle Scholar
  27. ——– (2014), “An evaluation of super-valid restricted randomization”, Journal of Agricultural, Biological and Environmental Statistics, 19, 472–480.MathSciNetCrossRefzbMATHGoogle Scholar
  28. Yates, F. (1933), “The formation of Latin squares for use in field experiments”, Empire Journal of Experimental Agriculture, 1, 235–244.Google Scholar

Copyright information

© International Biometric Society 2017

Authors and Affiliations

  1. 1.Statistical Consulting UnitAustralian National UniversityCanberraAustralia
  2. 2.Biostatistics Unit, Institute of Crop ScienceUniversity of HohenheimStuttgartGermany

Personalised recommendations